Elliptic Curves over Galois Number Fields

As E varies among elliptic curves defined over the rational numbers, a theorem of Bhargava and Shankar shows that the average rank of the Mordell-Weil group $E(\mathbb Q)$ is bounded. If we now fix a Galois number field K, how does the Mordell-Weil group E(K) behave on average as a Galois module? We will report on progress on this question, which is obtained by instead studying the associated p-Selmer groups of E/K as Galois modules. We construct some novel Selmer groups which describe certain invariants of these modules, and go on to study the behaviour of these new Selmer groups. This in turn allows us to give bounds for certain behaviour for the Mordell-Weil groups.